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Karnaugh Map Simplifier

Simplify small Boolean functions with 2, 3, and 4 variable Karnaugh maps. Enter minterms, add optional don't-care terms, or convert a Boolean expression into a K-map workflow.

DIG-003 focuses on Boolean-level SOP minimization. It does not perform FPGA mapping, propagation-delay optimization, power analysis, or physical gate placement.

Engineering tool

Karnaugh Map Simplifier

Simplify 2, 3, and 4 variable Boolean logic with Karnaugh maps, minterms, don't-care terms, and deterministic SOP grouping.

Calculator mode

V1 supports 2, 3, and 4 variable Karnaugh maps.

Enter comma-separated minterms, for example: 1,3,5,7 or Σm(1,3,5,7).

Optional comma-separated don't-care cells, for example: 0,2 or d(0,2).

Result console

Simplified SOP
C
Variables
3
Minterms
1, 3, 5, 7
Don't Cares
None
Groups
1
Verification
Equivalent

Karnaugh map grid

3-variable Karnaugh map
A \ BC00011110
00m01m11m30m2
10m41m51m70m6

Grouping

Karnaugh map grouping table
TermGroup SizeCells IncludedType
C4m1, m3, m5, m7Essential Prime Implicant

Formula reference

Karnaugh Map Simplification Rules

K-map simplification groups adjacent 1 cells in Gray-code order and creates a reduced sum-of-products expression.

Valid group sizes = 1, 2, 4, 8, 16K-map adjacency uses Gray codeSOP = ProductTerm1 + ProductTerm2 + ...A changing variable inside a group is removedDon't-care cells may expand groupsVerification: original truth table = simplified SOP truth table

Variable definitions

Minterm
truth-table row where output is 1
Don't-care
input condition that may be treated as 0 or 1
Prime implicant
group that cannot be expanded further
Essential prime implicant
group required to cover a unique minterm
SOP
sum of products form
Gray code
ordering where adjacent labels differ by one bit

Worked Examples

2-variable AND

Σm(3) simplifies to AB.

2-variable OR

Σm(1,2,3) simplifies to A + B.

XOR

Σm(1,2) remains A'B + AB' because no larger adjacent group covers both 1s.

3-variable majority

Σm(3,5,6,7) simplifies to AB + AC + BC.

4-variable wrap-around

Σm(0,2,8,10) uses edge wrapping and simplifies to B'D'.

Minterms 1,3,5,7

A full column group in a 3-variable map simplifies to C.

Don't-care grouping

Σm(1,3) with d(5,7) can simplify to C by using don't-care cells.

Expression simplification

(A & B) | (A & C) converts to minterms and verifies equivalent SOP output.

Fully minimized case

All minterms in a map simplify to logic 1.

No simplification possible

XOR-style isolated cells remain separate product terms.

Engineering Notes

  • Karnaugh maps are a visual Boolean minimization method for small logic functions.
  • Adjacent groups must contain 1, 2, 4, 8, or 16 cells.
  • K-map rows and columns use Gray-code order, not normal binary counting order.
  • Groups may wrap around the left and right edges or top and bottom edges.
  • Groups may overlap when that produces a smaller final SOP expression.
  • Don't-care cells may be used to enlarge groups, but they are not required output cells.
  • An essential prime implicant covers a required minterm that no other group covers.
  • A simpler SOP expression can reduce gate count, but it does not guarantee fastest hardware timing.

Common Mistakes

  • Using binary order instead of Gray-code order.
  • Forgetting wrap-around adjacency.
  • Treating don't-care terms as required 1 outputs.
  • Grouping cells in sizes that are not powers of two.
  • Missing overlapping groups that reduce the final expression.
  • Confusing SOP and POS forms.
  • Assuming the fewest Boolean terms always produce the fastest circuit.
  • Using K-map simplification as a substitute for timing analysis.

Support reference

FAQ

What is a Karnaugh map?

A Karnaugh map is a visual Boolean simplification method that arranges truth-table cells so adjacent cells differ by only one variable.

How does K-map simplification work?

Adjacent 1 cells are grouped in powers of two. Each group removes variables that change inside the group and produces a simpler SOP term.

How many variables can K-map handle?

This calculator supports 2, 3, and 4 variable Karnaugh maps. Larger maps are usually better handled with formal minimization tools.

What are minterms?

Minterms are truth-table row numbers where the Boolean function output is 1.

What are don't-care conditions?

Don't-care terms are input combinations that may be treated as either 0 or 1 to create larger groups and simpler logic.

What is a prime implicant?

A prime implicant is a group that cannot be expanded into a larger valid group while still covering only 1 or don't-care cells.

What is an essential prime implicant?

An essential prime implicant covers at least one required minterm that no other prime implicant covers.

Why does K-map use Gray code?

Gray-code ordering ensures adjacent cells differ by one variable, which is the condition required for valid Boolean grouping.

Can K-map simplify XOR logic?

K-map can represent XOR logic, but XOR often does not reduce well in SOP form because its 1 cells are not adjacent in a way that forms larger groups.

Does K-map optimize hardware timing?

No. K-map simplification reduces Boolean terms. It does not model propagation delay, fanout, hazards, routing, or physical implementation timing.

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